Generalization of Quantification for PLS Correlation

This study proposes a quantification algorithm for a PLS method with several sets of variables. We called the quantification method for PLS with more than 2 sets of data a generalization. The basis of the quantification for PLS method is singular value decomposition. To derive the form of singular value decomposition in the data with more than 2 sets more easily, we used the constraint, $a^ta+b^tb+c^tc=3$ not $a^ta=1$, $b^tb=1$, and $c^tc=1$, for instance, in the case of 3 data sets. However, to prove that there is no difference, we showed it by the use of 2 data sets case because it is very complicate to prove with 3 data sets. The keys of the study are how to form the singular value decomposition and how to get the coordinates for the plots of variables and observations.